Article
Determination of the Optimal Retention Level Based
on Different Measures
Başak Bulut Karageyik * and Şule Şahin
Department of Actuarial Sciences, Hacettepe University, 06800 Ankara, Turkey; [email protected]
* Correspondence: [email protected]; Tel.: +90-312-297-6160
Academic Editor: Michael McAleer
Received: 6 December 2016; Accepted: 18 January 2017; Published: 25 January 2017
Abstract: This paper deals with the optimal retention level under four competitive criteria: survival
probability, expected profit, variance and expected shortfall of the insurer’s risk. The aggregate claim
amounts are assumed to be distributed as compound Poisson, and the individual claim amounts
are distributed exponentially. We present an approach to determine the optimal retention level
that maximizes the expected profit and the survival probability, whereas minimizing the variance
and the expected shortfall of the insurer’s risk. In the decision making process, we concentrate on
multi-attribute decision making methods: the Technique for Order of Preference by Similarity to Ideal
Solution (TOPSIS) and the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) methods
with their extended versions. We also provide comprehensive analysis for the determination of the
optimal retention level under both the expected value and standard deviation premium principles.
Keywords: optimal reinsurance; survival probability; expected profit; expected shortfall; variance of
the insurer’s risk; TOPSIS; VIKOR
1. Introduction
There has been a growing interest in ruin probability, and considerable attention has been paid
to determine the optimal reinsurance level under the ruin probability constraint. As a very first
study, De Finetti discusses how optimal levels should be calculated for both the excess of loss and
proportional reinsurance under the minimum variance criterion for the insurer’s expected profit [1].
Dickson and Waters [2] develop De Finetti’s approach and focus on minimizing the ruin probability
instead of the variance criterion in [1]. Kaluszka presents the optimal reinsurance, which aims to
minimize the ruin probability for the truncated stop loss reinsurance [3]. Dickson and Waters consider
minimizing the ruin probability in compliance with a dynamic reinsurance strategy [4]. Kaishev and
Dimitrova suggest a joint survival optimal reinsurance model for the excess of loss reinsurance [5].
Nie et al. propose an approach to calculate the optimal reinsurance for a reinsurance arrangement in
the lower barrier model with capital injection [6]. Centeno and Simoes present a survey about the
state-of-the-art of optimal reinsurance [7].
In addition, Value at Risk (VaR) and Conditional Value at Risk (CVaR) are commonly-used risk
measures in the determination of optimal reinsurance. Borch proposes reinsurance as an effective
risk management tool for managing an insurer’s risk exposure [8]. Cai and Tan study the optimal
retention level according to the VaR and CVaR risk measures for a stop loss reinsurance [9]. Chi and
Tan suggest the optimal reinsurance model [10], which aims to minimize VaR and CVaR assuming that
the reinsurance premium principle satisfies three basic axioms: distribution invariance, risk loading
and stop-loss ordering preserving. Trufin et al. describe a VaR-type risk measure as the value at risk of
the maximal deficit of the ruin process in infinite time [11].
Previous studies indicate that researchers usually consider only a single constraint, such as ruin
probability, VaR, CVaR, expected profit or expected utility. Very few publications are available in the
J. Risk Financial Manag. 2017, 10, 4; doi:10.3390/jrfm10010004
www.mdpi.com/journal/jrfm
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literature that discuss the issue of optimal reinsurance under more than one constraint. For example,
Karageyik and Dickson [12] suggest optimal reinsurance criteria as the released capital, expected
profit and expected utility of resulting wealth under the minimum finite time ruin probability. They
aim to find the pair of initial surplus and reinsurance level that maximizes the output of these three
quantities under the minimum finite time ruin probability by using the translated gamma process to
approximate the compound Poisson process. They determine the optimal initial surplus and retention
level in a set of alternatives so that each pair satisfies the minimum ruin probability constraint. In the
decision making process, they use the Technique for Order of Preference by Similarity to Ideal Solution
(TOPSIS) method with the Mahalanobis distance.
Different from Karageyik and Dickson [12], in this study, we investigate the optimal retention level
that makes the survival probability and expected profit maximum, whereas variance and expected
shortfall minimum. We explore the survival probability in the determination of the optimal reinsurance
as a new criteria rather than a constraint. We also concentrate on the determination of the optimal
retention level in an alternative set that contains constant initial surplus and corresponding possible
reinsurance levels. The premium is calculated by using the expected value and standard deviation
premium principles. On the other hand, in the decision making process, we use two multi-attribute
decision making methods: TOPSIS and VlseKriterijumska Optimizacija I Kompromisno Resenje
(VIKOR), as well as their modified versions.
The paper is organized as follows. In Section 2, we introduce optimal reinsurance criteria to
determine the optimal retention level. We define the exact finite time ruin probability, expected profit,
variance and expected shortfall for a compound Poisson risk model from the insurer’s point of view
under the excess of loss reinsurance. The calculation of the expected profit is based on two fundamental
premium principles: the expected value and standard deviation. Since the main purpose of the paper
is to draw attention to determining the retention level on the excess of loss reinsurance, we show how
these criteria change according to the retention level, M. In Section 3 we present the decision theory
and Multi-Attribute Decision Making (MADM). In this context, we use two frequently-used MADM
methods: TOPSIS and VIKOR with their modified versions. In Section 4, we provide an analysis of
the determination of the optimal retention level by considering four criteria under the excess of loss
reinsurance. We show the effect of the premium principle assumption on the optimal reinsurance level.
Furthermore, we examine the sensitivity of the model to the choice of initial surplus and time horizons.
In Section 5, we conclude the paper.
2. Main Factors on the Determination of Optimal Reinsurance
The aim of this paper is to calculate the optimal retention levels by considering the survival
probability, expected profit, variance and expected shortfall criteria from the insurer’s point of view.
We show how these criteria are calculated under the excess of loss reinsurance arrangement.
2.1. The Exact Finite Time Ruin Probability
The surplus process (insurer’s risk process) comprises three main components: the initial capital
u, the premium income per unit of time and the aggregate claim amount up to time t, denoted by S(t).
Premium income is assumed as payable with a constant rate c per unit of time. The insurer’s surplus
(or risk) process, {U (t)}t≥0 , is defined by:
U (t) = u + ct − S(t).
The aggregate claim amount up to time t, S(t), is:
N (t)
S(t) =
∑
i =1
Xi ,
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where N (t) denotes the number of claims that occur in the fixed time interval [0, t]. The individual
claim amounts, modeled as independent and identically distributed (i.i.d.) random variables { Xi }i∞=1
with distribution function F ( x ) = Pr( X1 ≤ x ), such that F (0) = 0 and Xi is the amount of the i-th
claim. We use notations f and mk to represent the density function and k-th moment of X1 , respectively,
and it is assumed that c > E[ N ] m1 .
The ruin probability in finite time, ψ(u, t), is given by:
ψ(u, t) = Pr U (s) < 0
for some s,
0<s≤t ,
where ψ(u, t) is the probability that the insurer’s surplus falls below zero in the finite time interval (0, t].
In the classical risk model, the number of claims has a Poisson distribution with rate λ, so that
the aggregate claims have a compound Poisson process with Poisson parameter λ, and the individual
claim amounts are exponentially distributed with distribution function F ( x ). For a fixed value of t > 0,
the random variable S(t) has a compound Poisson distribution with Poisson parameter λt.
In practice, ψ(u, t), where t is the planning horizon of the company, is more interesting than
the infinite time ruin probability. The finite time ruin probability enables the insurance company to
develop the risk business or increase the premium if the risk business behaves badly. Especially in
non-life insurance, four of five years of finite time planning is reasonable [13]. Especially, the finite time
ruin probability may become useful and significant for the operation risk of the insurance company
when the real data are available.
The finite time ruin probability can be calculated analytically for a few special types of the
individual claim amount distribution. Prabhu proposes a finite time ruin probability formula [14],
when u ≥ 0, as a function of the distribution of the total claim amount in a specified time interval.
Seal develops Prabhu’s formula considering the exponential individual claims [15]. De Vylder suggests
a simple method that approximates a classical risk process {U (t)}t≥0 by another classical risk process
b (t)}t≥0 [16]. Segerdahl proposes a formula that extends the Cramer–Lundberg approximation by
{U
adding a time factor to obtain the finite time ruin probability [17]. Iglehart [18], Grandell [13] and
Asmussen and Albrecher [19] study the finite time ruin probability by using diffusion approximation
techniques. Dufresne et al. investigate the infinite time ruin probability when the aggregate claims
process is the standardized gamma process [20]. Then, Dickson and Waters suggest gamma and the
translated gamma process approximations in the classical risk model to calculate the finite time ruin
probability [21]. The finite time ruin probability can also be approximated by using Monte Carlo
simulations even though it is a time-consuming procedure.
Asmussen and Albrecher present an exact finite time ruin probability formula when the individual
claim amounts are exponentially distributed [19]. In this formula, it is assumed that the individual
claim amounts are distributed exponentially with parameter β with β = 1, the number of claims have a
Poisson distribution with the parameter λ and the premium rate per unit of time is equal to one (c = 1).
Then, the finite time ruin probability is calculated as:
ψ(u, t) = λ exp{−(1 − λ)u} −
where:
1
π
Z π
f1 (x) f2 (x)
0
f3 (x)
dx,
(1)
√
√
f 1 ( x ) = λ exp 2 λt cos( x ) − (1 + λ)t + u( λ cos( x ) − 1) ,
√
√
f 2 ( x ) = cos u λ sin( x ) − cos u λ sin( x ) + 2x ,
and:
√
f 3 ( x ) = 1 + λ − 2 λ cos( x ).
The major drawback of this approach is the limitation of the parameter of the individual claims
distribution (β = 1) and premium rate (c = 1). When β 6= 1, the following equation is applied [19].
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ψλ,β (u, t) = ψ λ ,1 ( βu, βt),
(2)
β
and the following equation is applicable when c 6= 1 [22].
ψλ,c (u, t) = ψ λ ,1 (u, ct).
(3)
c
The Exact Finite Time Ruin Probability on the Excess of Loss Reinsurance
Under the excess of loss reinsurance, a claim is shared between an insurer and a reinsurer
according to the fixed amount called the retention level, M. When a claim X occurs, the insurer
pays X I = min( X, M ), and the reinsurer pays XR = max(0, X − M ) with X = X I + XR . Hence, the
distribution function of X I , FX I ( x ), is:
(
FX I ( x ) =
FX ( x )
1
x<M,
x ≥ M,
for
for
and the moments of X I are:
n
E[( X I ) ] =
ZM
x n f ( x )dx + Mn 1 − F ( M ) .
(4)
0
Similarly, the moments of XR are:
n
E[( XR ) ] =
Z∞
x−M
n
f ( x )dx.
(5)
M
Under the excess of loss reinsurance, the aggregate claims for the reinsurer have a compound
Poisson distribution with Poisson parameter λexp(− βM ), and the individual claim amounts are
exponential distributed with parameter β. In a similar manner, the aggregate claims for the insurer have
a compound Poisson parameter λ(1 − exp(− βM)), and the individual claim amounts are exponentially
distributed with parameter β [23].
The survival probability is defined as the probability that ruin does not occur in the finite time
horizon (0, t] and is shown as ψ(u, t) = (1 − ψ(u, t)). In this study, we aim to determine the optimal
retention level that makes the insurer’s survival probability maximum.
2.2. Variance of the Insurer’s Risk
When the individual claim amount is distributed exponentially with parameter β, the moments
of the insurer’s individual claim amount under the excess of loss reinsurance can be obtained as [24]:
mk = E[ X kI ] =
k
γ(k, βM )
βk
for
k = 1, 2, · · · ,
(6)
where γ(k, M) is the incomplete gamma function and defined as:
γ(k, M ) =
Z M
0
tk−1 e−t dt.
The variance of the individual claim amount for the insurer’s risk is calculated by using the
variance principle.
V ( X I ) = E[ X 2I ] − E[ X I ]2 ,
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and then by using the assumption of the classical risk model, the variance of the aggregate total claim
amount for the insurer, denoted as V (S I ), is calculated as [23],
V ( S I ) = E [ X I ]2 V ( N ) + E [ N ]V ( X I ).
where E[ N ] is the expected number of claims and V ( N ) is the variance of the number of claims.
In this study, we aim to determine the optimal retention level that makes the variance of the insurer’s
risk minimum.
2.3. Expected Profit
In general, the expected profit of the insurance company is determined as the difference between
the insurer’s income and liabilities to the policyholders. Insurer’s income is the total premium income,
whereas liabilities are the benefit payments. Insurer’s profit is influenced by many factors, such as
pure risk premium, total claim amount, reinsurance level, insurance and reinsurance loading factor,
investment incomes, taxes, capital gains and dividends. In this study, we assume that the premiums
and claims are the main components of the insurance profit. Thus, we ignore the other factors, such as
investment incomes, dividend payments and taxes.
Premium principles have a significant influence on the calculation of the expected profit. In this
study, we calculate the expected profit according to two basic premium principles. First, we use the
expected value premium principle, which is commonly used in the literature. This premium principle
depends on the expected aggregate claims, insurance and reinsurance loading factors. The second
method is the standard deviation premium principle, which considers the expected value, as well as
the standard deviation of the aggregate claims.
In this study, we aim to calculate the optimal retention level that makes the insurer’s expected
profit maximum by using the expected and the standard deviation premium principles.
2.3.1. Expected Value Premium Principle
In the classical risk model, it is assumed that the number of claims has a Poisson distribution
with parameter λ. According to the expected value premium principle with the insurance loading
factor θ and the reinsurance loading factor ξ, the insurer’s premium income per unit of time after the
reinsurance premium (i.e., net of reinsurance) is defined as:
c∗
= Total Premium Income - Reinsurance Premium,
= (1 + θ ) E [ S ] − (1 + ξ ) E [ S R ],
= (1 + θ ) E [ N ] E [ X ] − (1 + ξ ) E [ N ] E [ X R ],
where E[S] is the expected aggregate claim and E[SR ] is the expected aggregate claim paid by the
reinsurer. It is also assumed that ξ ≥ θ > 0 and that c∗ > λ E[ X I ].
The net profit of the insurance company after the reinsurance arrangement is obtained by
subtracting the expected total claim amount paid by the insurer, E[S I ], from the expected net insurance
premium income, c∗ .
Net Insurance Profit = c∗ − E[S I ].
where E[S I ] = E[ N ] E[ X I ].
2.3.2. Standard Deviation Premium Principle
According to the standard deviation premium principle with loading α, the insurer’s premium
income per unit of time after the reinsurance premium (i.e., net of reinsurance) is defined as:
c∗ =
E[S] + α
q
q
V (S) − E[SR ] + α V (SR ) ,
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where V (SR ) denotes the variance of the aggregate claim amount paid by the reinsurance. This method
is preferred when the fluctuation of the aggregate claims is important. Hence, this method enables the
insurer to calculate a more accurate premium than the expected value premium principle provides.
2.4. Expected Shortfall
Value at Risk (VaR) is the probability that the loss on the portfolio over the given time horizon
exceeds a threshold value. VaR of a portfolio at a confidence level p ∈ (0, 1) is given by the smallest
number l, such that the probability of the loss L does not exceed l is at least (p) [25]. L is the loss of a
portfolio, and it is usually appropriate to assume in insurance contexts that the loss L is non-negative.
VaR p ( L) is the level p-quantile, i.e.:
VaR p ( L) = min{l ∈ R : Pr [ L ≤ l ] ≥ p}.
Expected Shortfall (ES) is one of the financial risk measures to investigate the market risk or
credit risk of a portfolio. Expected Shortfall is defined as an average of VaR p of X at level p. Expected
Shortfall is preferred to VaR, since it is more sensitive to the shape of the loss distribution in the tail of
the distribution. Expected Shortfall is also called CVaR, Average Value at Risk (AVaR) or Expected Tail
Loss (ETL). The ES at confidence level p ∈ (0, 1) is given by the following equation:
ES p ( X ) =
1
1− p
Z1
VaRu ( X )du.
p
The aggregate claims for the insurer have a compound Poisson parameter λ(1 − exp(− βM )),
and the individual claim amounts are exponentially distributed with parameter β under the excess
of loss reinsurance. We calculate the ES for the compound Poisson distribution according to the
retention level, M. The ES of a compound Poisson model is calculated by using the R programming
language [26].
An increase in the retention level causes an increase of the insurer’s responsibility, and thus, the
ES increases. In this study, we aim to determine the optimal retention level that makes the insurer’s ES
minimum.
3. Multi-Attribute Decision Making
Multiple-Criterion Decision Making (MCDM) is used to make a decision in the presence of
multiple, usually conflicting criteria. The problems of MCDM are constitutively classified into two
categories: MADM and Multiple Objective Decision Making (MODM). Multiple Objective Decision
Making depends on designing a problem, whereas MADM is based on solving a problem by selection
among a finite number of alternatives [27]. Hence, we focus on MADM to determine the optimal
retention levels.
Multi-Attribute Decision Making methods are mainly comprised of four components: alternatives,
attributes, weight of the relative importance of each attribute and measures of the performance of
alternatives regarding the attributes. The decision matrix D is an m × n matrix, and it shows m
alternative options, which need to be assessed on n attributes (criteria). Each element, Xij , is either a
single numerical value or a single grade, representing the performance of alternative i on criterion
j. The decision table is shown in Table 1. In our analysis, we assume that the variance of the insurer
risk, expected shortfall, expected profit and survival probability are the attributes (criteria), whereas
the alternative sets are retention levels. Hence, Xij shows the values of each criterion according to the
corresponding retention levels.
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Table 1. Decision matrix for Multi-Attribute Decision Making (MADM) methods.
Attributes (Criteria) (Cj )
Alternatives (Ai )
A1
A2
A3
..
.
Am
C1
C2
C3
···
Cn
X11
X21
X31
..
.
Xm1
X12
X22
X32
..
.
Xm2
X13
X23
X33
..
.
Xm3
···
···
···
..
.
···
X1n
X2n
X3n
..
.
Xmn
In MADM, the importance of each attribute is described by the weights of the attributes. A set of
weights for n attributes is shown as:
w T = ( w1 , w2 , · · · , w n ) ,
n
where ∑ w j = 1.
j =1
Hwang and Yoon suggest four techniques in MCDM to calculate the weights of criteria [28]:
the eigenvector method, the weighted least squares method, the entropy method and the linear
programming technique for multidimensional analysis of the preference method.
In this study, we choose to focus on the entropy method of the determination of the weights of
the criteria due to its practicality. In the entropy method, it is assumed that a criterion for the amount
of uncertainty is presented by a discrete probability distribution, Pi . The project outcomes of attribute
j, Pij can be defined as:
Xij
Pij = m
,
∀ i,j.
∑ Xij
i =1
The entropy Ej of the set of project outcomes of attribute j is:
m
Ej = −k ∑ Pij ln Pij ,
∀ i,j.,
i =1
where k = 1/ ln(m) and 0 ≤ Ej ≤ 1. The degree of diversification, d j , is calculated as d j = 1 − Ej for
all j and then, the weights are calculated as:
wj =
dj
n
,
∀ j.
∑ dj
j =1
In the expression of the inter-attribute preference information, the methods for the cardinal
preference of the attribute given are commonly preferred. The methods for the cardinal preference of
the attribute given can be categorized into seven methods: the Linear Assignment Method (LAM), the
Simple Additive Weighting Method (SAW), the Hierarchical Additive Weighting Method (HAWM),
the Analytical Hierarchy Process (AHP), the Elimination and Choice Translating Reality (ELECTRE),
the TOPSIS and the VIKOR.
3.1. Technique for Order of Preference by Similarity to Ideal Solution
3.1.1. Technique for Order of Preference by Similarity to Ideal Solution Method with
Euclidean Distance
Hwang and Yoon propose the TOPSIS method to determine the best alternative based on the
concept of a compromised solution [28]. This method is based on choosing a solution with the shortest
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Euclidean distance from the positive ideal solution and the farthest Euclidean distance from the
negative ideal solution. One of the main assumption of the TOPSIS method is that the criteria are
monotonically increasing or decreasing. The ideal solution is defined as the one that maximizes the
benefit criteria and minimizes the cost criteria, whereas the negative ideal solution is defined as the
one that maximizes the cost criteria and minimizes the benefit criteria. The ranking of the alternatives
is calculated according to the relative proximity to the ideal solution.
References to the TOPSIS method for real data can be found in a number of studies, including
Wang and Hsu [29], Wu and Olson [30], Shih et al. [31], Jahanshahloo et al. [32], Bulgurcu [33],
Zhu et al. [34] and Hosseini et al. [35].
The procedures of the TOPSIS are described as follows:
Step 1: The decision matrix is normalized by using the vector-normalization technique.
rij = s
xij
,
m
∑ (xij )
2
i =1
where rij is the normalized value for i = 1, 2, · · · , m and j = 1, 2, · · · , n.
Step 2: Weighted-normalized values are calculated by using the weight vector ω = (ω1 , ω2 , · · · , ωn ).
Vij ( x ) = w j rij ,
i = 1, · · · , m and j = 1, · · · , n.
Step 3: The positive ideal points S+ are determined as:
S+
=
n
+
S1+ , S2+ , · · · , S+
j , · · · , Sn
o
= {(max Vij | j ∈ J ), (min Vij | j ∈ J 0 ) | i = 1, 2 · · · , m}.
i
i
The negative ideal points S− are determined as:
S−
=
n
−
S1− , S2− , · · · , S−
j , · · · , Sn
o
= {(min Vij | j ∈ J ), (max Vij | j ∈ J 0 ) | i = 1, 2 · · · , m}.
i
i
where J = { j = 1, 2, · · · , n| j associated with the benefit criteria} and
0
J = { j = 1, 2, · · · , n| j associated with the cost criteria}.
Step 4: The distance between each alternative and positive ideal solution is calculated by using
n-dimensional Euclidean distance.
v
u n
u
+
2
di = t ∑ (Vij − S+
j ) ,
i = 1, 2, · · · , m.
j =1
The distance between each alternative and negative ideal solution is calculated by using
n-dimensional Euclidean distance.
v
u n
u
−
2
i = 1, 2, · · · , m.
di = t ∑ (Vij − S−
j ) ,
j =1
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Step 5: The relative closeness of each alternative to the ideal solution is calculated as:
Ci =
di−
di+ + di−
,
where Ci ∈ [0, 1] for i = 1, · · · , m. The results are sorted according to the value of Ci . A higher Ci
means that Ai is a better solution.
3.1.2. Technique for Order of Preference by Similarity to Ideal Solution Method with
Mahalanobis Distance
The Euclidean distance measure depends on the ordinary (i.e., straight-line) distance between
two points in Euclidean space. Hence, the TOPSIS method with Euclidean distance assumes that
there is no relationship between the attributes. This approach suffers from information overlap and
either overestimates or underestimates the attributes that take slack information [36]. When attributes
are dependent and influence each other, the application of TOPSIS based on Euclidean distances can
lead to inaccurate estimation of the relative significances of alternatives and cause improper ranking
results [37]. Therefore, the Mahalanobis distance measure technique is suggested as an alternative to
the Euclidean distance in the TOPSIS method.
The Mahalanobis distance, also called quadratic distance, was introduced by Mahalanobis [38].
For a multivariate vector x = ( x1 , x2 , x3 , . . . , x N ) T from a group of observations with mean
µ = (µ1 , µ2 , µ3 , . . . , µ N ) T and the covariance matrix Σ, the Mahalanobis distance is defined as follows.
DM (x) =
q
(x − µ ) T Σ −1 (x − µ ).
The Mahalanobis distance standardizes via the factor of the inverse of the covariance matrix
When the attributes are not related to one another, the weighted Mahalanobis distance and the
weighted Euclidean distance will be equivalent [36].
References to the TOPSIS method with Mahalanobis distance can be found in a number of studies,
such as Wang and Wang [36], Garca and Ibarra [39], Chang et al. [40] and Lahby et al. [41].
The TOPSIS method with Mahalanobis distance has the following steps. The decision matrix
consists of the element of Xij and is standardized as follows.
Σ −1 .
rij = s
xij
m
.
∑ xij
i =1
Let the ideal solution and negative ideal solution (anti-ideal solution) be S+ and S− , respectively,
as in the case of TOPSIS, and Ai denote the i-th alternative. Hence, the Mahalanobis distance from Ai
to the positive/negative ideal solution point is calculated as:
d (ri , S + / − ) =
q
/−
/−
{S+
− rij } T Ω T Σ−1 Ω{S+
− rij }
j
j
i = 1, 2, · · · , m
where ω is the weight vector, such as ω=(ω1 , ω2 , · · · , ωn ), and Ω is defined as
√
√
√
Ω = diag( w1 , w2 , · · · , wn ). The closeness of each alternative is calculated as:
ci =
d (ri , S − )
d (ri , S − ) + d (ri , S + )
i = 1, 2, · · · , m .
The results for the alternatives are sorted according to the value of ci . Higher ci suggests that Ai
is a better solution.
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3.1.3. Modified TOPSIS Method
The traditional TOPSIS method uses Euclidean distance to calculate the overall performance of
each alternative. In addition, this method compares all alternatives with the ideal and negative ideal
points of each criterion. Deng et al. propose the weighted Euclidean distance instead of creating a
weighted decision matrix [42]. Thus, positive and negative ideal solutions are not dependent on the
weighted decision matrix [27]. According to the modified TOPSIS method, the positive ideal solution
(R+ ) is defined as:
R
+
=
=
max
min
( ∑ rij /j ∈ J ), ( ∑ rij /j ∈ J )/i = 1, 2, · · · , m
i
+
R1 , R2+ , R3+ , · · ·
∗
i
+
, Rm
and the negative ideal solution (R− ) is defined as:
R
−
=
=
min
max
i
i
( ∑ rij /j ∈ J ), ( ∑ rij /j ∈ J )/i = 1, 2, · · · , m
∗
R1− , R2− , R3− , · · · , R−
m
where J = ( j = 1, 2, · · · , n)/j is associated with beneficial attributes and J ∗ = ( j = 1, 2, · · · , n)/j is
associated with non-beneficial attributes. The weighted Euclidean distance is calculated as:
n
/− 2 0.5
Di+/− = { ∑ w j (rij − R+
) } ,
j
i = 1, 2, · · · , m.
j =1
The relative closeness of a particular alternative to the ideal solution is expressed as follows:
Ci =
Di−
Di+ + Di−
,
where Ci ∈ [0, 1] for i = 1, · · · , m. The set of alternatives is sorted in increasing order; the maximum
value of Ci is the most preferable and feasible solution.
3.2. VIseKriterijumska Optimizacija I Kompromisno Resenje
The VIKOR method, or multi-criteria optimization and compromise solution, is developed for the
optimization of complex MCDM systems. It focuses on ranking and selecting from a set of alternatives
in the presence of conflicting and incommensurable criteria. This method provides a multiple-criteria
ranking index based on the particular measure of “closeness to the ideal” solution [43]. The idea of the
compromise solution has been introduced in MCDM by Po-Lung Yu [44] and Milan Zeleny [45] and
developed by Opricovic and Tzeng [46–49] and Tzeng et al. [50–52].
The VIKOR method depends on the L p -metric.
L p,i =
m
∑ (w j [(Xij )max − (xij )]/[(xij )max − (xij )min ]) p
1/p
,
j =1
with 1 ≤ p ≤ ∞ and i = 1, 2, · · · , n.
following steps:
The compromise ranking algorithm VIKOR has the
Step 1: Determine the best X j∗ and the worst X j− values of all criterion functions, j = 1, 2, · · · , n. If the
j-th function represents a benefit criteria, X j∗ = max Xij (aspired/desired level), whereas if the j-th
i
function represents a cost criteria, X j∗ = min Xij (non-aspired/undesired level).
i
J. Risk Financial Manag. 2017, 10, 4
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Step 2: Compute the values Si (the maximum group utility) and Ri (the minimum individual regret of
the opponent) for i = 1, 2, · · · , m by the relations:
Si = L1,i =
m
∑
w j [ X j∗
− xij ]/[ X j∗
j =1
and:
− X j− ]
,
Ri = max w j [ X j∗ − xij ]/[ X j∗ − X j− ] ,
where w j is the weight of the j-th criterion, which express the relative importance of the criteria.
Step 3: Compute the synthesized index, Qi for i = 1, 2, · · · , m;
Qi = ν(Si − S∗ )/(S− − S∗ ) + (1 − ν)( R j − R∗ )/( R− − R∗ ),
where S∗ = min Si , S− = max Si , R∗ = min Ri , R− = max Ri and ν is introduced as the weight of the
i
i
i
i
strategy of Si and Ri .
Step 4: Rank the alternatives, and sort by the values of S, R and Q in decreasing order. The results are
three ranking lists.
0
Step 5: Propose as a compromise solution the alternative ( a ), which is ranked the best by the measure
minimum Q if the following two conditions are satisfied:
00
0
00
C1. “Acceptable advantage”: Q( a )-Q( a ) ≤ DQ where a is the alternative with the second
position in the ranking list by Q, DQ = 1/(n − 1) and J is the number of alternatives.
0
C2. “Acceptable stability in decision making”: Alternative a must also be the best ranked by
S or/and R. This compromise solution is stable within a decision making process, which could be:
“voting by majority rule” (when ν > 0.5 is needed), “by consensus” ν ≈ 0.5 or “with vote” (ν < 0.5).
Here, ν is the weight of the decision-making strategy “the majority of criteria” (or “the maximum
group utility”).
If one of the conditions is not satisfied, a set of compromise solutions is proposed, which
consists of:
0
00
•
Alternatives a and a if only condition C2 is not satisfied or
•
Alternatives a , a ,· · · , an if condition C1 is not satisfied; and an is determined by the relation
0
Q( an ) − Q( a ) < DQ for maximum n (the positions of these alternatives are “in closeness”).
0
00
The best alternative, ranked by Q, is the one with the minimum value of Q [53].
Huang et al. propose a revised VIKOR method that depends on the normalization of the levels of
regret [53]. The revised VIKOR method enables one to choose the best alternative by using the best
and the worst values of each criterion instead of only the best value of each criterion.
3.3. A Revised VIseKriterijumska Optimizacija I Kompromisno Resenje Method
Huang et al. revise the VIKOR method by using the regret theory [53]. The VIKOR method
describes the regret as the discrimination between alternatives and the best value of each criterion,
whereas regret theory defines the regret as the choiceless utility. In the revised VIKOR method, Si
(choiceless utilities) and Ri (discontent utilities) can be defined as:
J. Risk Financial Manag. 2017, 10, 4
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f (x) =




n
m
1
Si = ∑ w j n −
∑ ( Xkj − Xij ) ,
1
j =1
k =1
i f Xij < Xkj
p


0,
otherwise
and:
n
∗
Ri = ∑ w j ( X j − Xij ) ,
j =1
p
X j∗
where
is the best value of the j-th criterion and k.k p denotes the L p -norm. In this paper, we use the
L2 -norm to obtain the values of S and R. Then, the synthesized index Q is calculated as:
Qi = ν(Si − S∗ )/(S− − S∗ ) + (1 − ν)(( R j ) − ( R∗ ))/(( R− − ( R∗ )),
where S∗ = min Si , S− = max Si , R∗ = min Ri , R− = max Ri and ν is introduced as the weight of the
i
i
i
i
strategy of Si and Ri .
4. Analysis of the Optimal Retention Level
In this section, we aim to calculate the optimal retention level M for the constant initial surplus
under the excess of loss reinsurance. We determine the optimal retention level that makes the survival
probability and expected profit maximum, whereas the variance and the expected shortfall of the
insurer risk minimum.
We consider that the individual claim has an exponential distribution with parameter β = 1,
and the number of claims has a Poisson distribution with parameter λ = 1. We use two different
premium principles: the expected value and the standard deviation. It is assumed that the insurance
loading factor is 0.1, and the reinsurance loading factor is 0.15 in the expected value premium principle,
whereas the loading is assumed as 0.01 in the standard deviation premium principle. In the decision
process, we use the TOPSIS and VIKOR methods with their improved cases. We investigate and
compare the optimal retention level according to these decision making process techniques.
4.1. The Effect of Criteria on the Optimal Retention Level under the Expected Value Premium Principle
We design a set of alternatives that comprise initial surplus and retention level. We assume that
the initial surplus is constant, such as 1, 5, 10. Dickson shows the condition for the minimum retention
level as M > log(ξ/θ ) [23]. This condition is obtained by applying c > λE[ X I ] to the exponential
individual claim amount with parameter β = 1.
In particular, we consider values of retention level ( M) starting from 0.4055 (log(0.15/0.1)) and
increasing by 0.1 to three-times bigger than the initial surplus. There is no change in the results even if
the increases are made with smaller increments, such as 0.05. Thus, under each fixed initial capital, all
alternative retention levels are obtained. Then, we calculate the variance of the insurer risk, expected
shortfall, expected profit and survival probability by using fixed initial surplus and each corresponding
retention level. Therefore, we obtain an outcome set that presents all possible combinations of the
retention level and the corresponding value of each criterion.
4.1.1. Determination of the Normalized Weights
We measure the importance of the criteria according to different initial surpluses and time horizons
under the entropy method. Table 2 presents the weights of the variance (Var ( X I )), Expected Shortfall
(ES0.95 ), Expected Profit (EP) and survival probability (ψ(u, t)) for the insurer risk by using the entropy
method under the expected value premium principle. The results indicate that the variance has a
dominant impact among other criteria. If these weights are used in the TOPSIS or VIKOR methods,
the optimal values will be affected by the weight of the variance criterion. Thus, the optimal levels will
J. Risk Financial Manag. 2017, 10, 4
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be determined at the point where the variance criteria have the maximum value. The effects of the
other criteria will not be observed. Since we aim to get minimum distortion on the weights, the criteria
are assigned to equal weights. Hence, we carry out the following assumption:
w1 = w2 = w3 = w4 = 0.25.
Table 2. The weights of criteria for the entropy method under the expected value premium principle.
The Weights of Criteria under the Expected Value Premium Principle
u=1
u=5
u = 10
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
Var ( X I )
ES0.95
EP
ψ(u, t )
0.7028
0.6999
0.6983
0.6962
0.6943
0.6891
0.6857
0.6833
0.7407
0.7339
0.7241
0.7274
0.7276
0.7174
0.7131
0.7088
0.7394
0.7324
0.7222
0.7263
0.7268
0.7172
0.7132
0.7088
0.0202
0.0241
0.0264
0.0294
0.0321
0.0395
0.0439
0.0473
0.0173
0.0262
0.03886
0.03412
0.0336
0.0456
0.0505
0.0557
0.0175
0.0268
0.0404
0.0349
0.0343
0.0469
0.0519
0.0574
0.0002
0.0002
0.0002
0.0001
0.0001
0.00001
0.00009
0.00025
0.00004
0.0002
0.0005
0.0008
0.0011
0.0026
0.0035
0.0039
0.00000003
0.0000005
0.000003
0.000009
0.00002
0.0002
0.0005
0.0009
0.2768
0.2757
0.2751
0.2742
0.2735
0.2714
0.2701
0.2691
0.2420
0.2398
0.2366
0.2376
0.2377
0.2344
0.2330
0.2316
0.2431
0.2408
0.2374
0.2387
0.2389
0.2357
0.2344
0.2330
ES: Expected Shortfall; EP: Expected Profit.
4.1.2. Evaluation of Criteria under Multi-Attribute Decision Making for the Expected Value
Premium Principle
The optimal retention level is obtained by using five different MADM methods: TOPSIS, Modified
TOPSIS (M-TOPSIS), TOPSIS with Mahalanobis distance (TOPSIS-Mahalanobis), VIKOR and Revised
VIKOR (R-VIKOR) methods. The rankings, when initial surplus is equal to five and the time horizon is
one, are obtained as shown in Figure 1.
It is observed that for the TOPSIS-Euclidean (TOPSIS-E), M-TOPSIS and TOPSIS-Mahalanobis
methods, the best alternative is determined by the point at which the closeness index is maximum.
However, in the VIKOR and R-VIKOR methods, the best alternative is determined by the point at
which the synthesize index is maximum.
J. Risk Financial Manag. 2017, 10, 4
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8.405
0.6
0.5
0.405
Retention Levels
8.405
0.405
Retention Levels
8.405
Retention Levels
R−VIKOR
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
Synthesized Index
0.8
1.0
1.0
VIKOR
Synthesized Index
0.7
Closeness Index
0.45
0.405
0.8
0.9
0.65
0.60
0.55
0.50
Closeness Index
0.60
0.55
0.50
0.45
Closeness Index
TOPSIS−Mahalanobis
1.0
M−TOPSIS
0.65
TOPSIS
0.405
4.405
8.405
12.405
Retention Levels
0.405
4.405
8.405
12.405
Retention Levels
Figure 1. Comparison of rankings under the expected value premium principle when u = 5 and t = 1.
TOPSIS: Technique for Order of Preference by Similarity to Ideal Solution; M-TOPSIS: Modified TOPSIS;
VIKOR: VIseKriterijumska Optimizacija I Kompromisno Resenje; R-VIKOR: Revised VIKOR.
Since the revised VIKOR method takes the variation between each alternative and the best value
of the criterion into account, this method is insufficient in determining the optimal retention level.
We compare optimal retention levels for different initial surpluses and time horizons to observe the
changes on rankings for five methods. Table 3 presents the optimal retention levels for different
initial surplus and time horizons under the expected value premium principle. The results show
that the optimal retention levels do not differ significantly for all methods, except the revised VIKOR
method. The TOPSIS method with Euclidean distance gives the same results as the modified TOPSIS
method for each initial surplus and time horizon. The findings are quite surprising and suggest that
the VIKOR method gives closer results to the TOPSIS-Mahalanobis more than the TOPSIS-E and
M-TOPSIS methods. An important implication of these findings is that the optimal retention levels are
not influenced by the changes on initial surpluses and time horizons. In the revised VIKOR method,
the optimum levels are determined as the highest retention level in each alternative set.
J. Risk Financial Manag. 2017, 10, 4
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Table 3. Optimal retention level under the expected value premium principle.
Optimal Retention Levels
u=1
u=5
u = 10
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
TOPSIS-E
M-TOPSIS
TOPSIS-Mahalanobis
VIKOR
R-VIKOR
1.1055
1.0055
1.0055
1.1055
1.1055
1.0055
1.1055
1.4055
1.3055
1.4055
1.3055
1.3055
1.2055
1.2055
1.4055
1.3055
1.4055
1.3055
1.3055
1.2055
1.3055
1.1055
1.0055
1.0055
1.1055
1.1055
1.0055
1.1055
1.4055
1.3055
1.4055
1.3055
1.3055
1.2055
1.2055
1.4055
1.3055
1.4055
1.3055
1.3055
1.2055
1.3055
1.1055
1.2055
1.2055
1.2055
1.1055
1.1055
1.1055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.2055
1.1055
1.0055
1.0055
1.0055
1.0055
1.0055
1.0055
1.2055
1.1055
1.1055
1.1055
1.1055
1.1055
1.0055
1.2055
1.1055
1.1055
1.2055
1.2055
1.1055
1.1055
2.8055
2.9055
2.9055
2.9055
2.9055
2.9055
2.9055
14.906
14.906
14.906
14.906
14.906
14.906
14.906
29.806
29.806
29.806
29.906
29.906
29.906
29.906
E: Euclidean; M: Modified; R: Revised.
The covariance matrices of the normalized attributes for t = 1, 5, 10 are given in Table 4. When the
covariance matrix is equal to the identity matrix, the Mahalanobis distance turns into the Euclidean
distance. In addition, when the covariance matrix is diagonal, the Mahalanobis distance can be shown
as the normalized Euclidean distance [36]. The small covariances between each pair of criteria indicate
that there is no relationship between the criteria. Thus, the optimal levels obtained by the Mahalanobis
distance measure are very close to the values obtained by the Euclidean distance measure.
Table 4. Covariance matrices of normalized attributes for t = 1, 5, 10.
Covariance Matrices
t=1
Var ( X I )
ES p
EP
ψ(u, t)
Var ( X I )
1.41 × 10−4
2.01×10−5
−2.02 × 10−8
7.93×10−5
ES0.95
2.01 × 10−5
4.64 × 10−6
−3.45 × 10−9
1.44×10−5
EP
−2.02×10−8
−3.45 × 10−9
3.12×10−12
−1.26 × 10−8
ψ(u, t)
7.93×10−5
1.44 × 10−5
−1.26 × 10−8
5.11×10−5
t=5
Var ( X I )
ES0.95
EP
ψ(u, t)
Var ( X I )
1.41×10−4
3.15×10−5
−6.15×10−7
7.93×10−5
ES0.95
3.15×10−5
9.02×10−6
−1.59×10−7
2.12×10−5
EP
−6.15×10−7
−1.59×10−7
2.93×10−9
−3.86×10−7
ψ(u, t)
7.93×10−5
2.12×10−5
−3.86×10−7
5.11×10−5
t = 10
Var ( X I )
ES0.95
EP
ψ(u, t)
Var ( X I )
1.41×10−4
3.82×10−5
−1.97×10−6
7.93×10−5
ES0.95
3.82×10−5
1.23×10−5
−6.13×10−7
2.50×10−5
EP
−1.97×10−6
−6.13×10−7
3.08×10−8
−1.25×10−6
ψ(u, t)
7.93×10−5
2.50×10−5
−1.25×10−6
5.11×10−5
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4.2. The Effect of Criteria on the Optimal Retention Level under the Standard Deviation Premium Principle
4.2.1. Determination of the Normalized Weights
Table 5 shows the weights of the criteria according to the entropy method under the standard
deviation premium principle.
Table 5. The weights of the criteria for the entropy method under the standard deviation
premium principle.
The Weights of Criteria under the Standard Deviation Premium Principle
Var ( X I )
ES0.95
EP
ψ(u, t )
u=1
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
0.7936
0.7869
0.7825
0.7780
0.7742
0.7641
0.7588
0.7551
0.0228
0.0271
0.0296
0.0329
0.0358
0.0438
0.0486
0.0523
0.0030
0.0069
0.0099
0.0121
0.0138
0.0183
0.0200
0.0209
0.1805
0.1790
0.1780
0.1770
0.1761
0.1738
0.1726
0.1718
u=5
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
0.7102
0.7040
0.6950
0.6980
0.6980
0.6875
0.6822
0.6770
0.0166
0.0251
0.0373
0.0327
0.0322
0.0437
0.0483
0.0532
0.0000
0.0001
0.0004
0.0008
0.0013
0.0043
0.0072
0.0094
0.2731
0.2707
0.2673
0.2684
0.2684
0.2644
0.2623
0.2604
u = 10
t=1
t=2
t=3
t=4
t=5
t = 10
t = 15
t = 20
0.6976
0.6914
0.6822
0.6859
0.6863
0.6777
0.6740
0.6697
0.0165
0.0253
0.0382
0.0330
0.0324
0.0443
0.0490
0.0542
0.0000
0.0000
0.0000
0.0000
0.0000
0.0003
0.0008
0.0016
0.2859
0.2833
0.2796
0.2811
0.2812
0.2777
0.2762
0.2744
As is seen in Table 5, variance criteria have the highest weight, and thus, they have the biggest
effect on determining the optimal retention level. This weighting method might causes optimal values
to be obtained at the points where the variance is maximum. It is similar to the expected value premium
case. Thus, we assume that the weights of the criteria are equal.
4.2.2. Evaluation of Criteria under Multi-Attribute Decision Making for the Standard Deviation
Premium Principle
When initial surplus is equal to five and the time horizon is one, the rankings are obtained
under the standard deviation premium principle as shown in Figure 2. The same alternative is
obtained as the optimal solution for all methods except the R-VIKOR method. The M-TOPSIS and
TOPSIS-Mahalanobis depend on the closeness index, which gives the best solution as the highest
index. However, the VIKOR method uses the smallest value in the synthesized index for the best
solution. The optimal retention level for different initial surpluses and time horizons under the
standard deviation premium principle is given in Table 6.
J. Risk Financial Manag. 2017, 10, 4
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M−TOPSIS
TOPSIS−Mahalanobis
0.405
8.405
0.405
Retention Levels
0.60
0.55
8.405
0.405
Retention Levels
8.405
Retention Levels
R−VIKOR
0.8
0.6
0.4
0.2
0.0
0.4
0.6
Synthesized Index
0.8
1.0
1.0
VIKOR
Synthesized Index
0.50
Closeness Index
0.55
0.50
0.45
Closeness Index
0.55
0.50
0.45
Closeness Index
0.65
TOPSIS
0.405
4.405
8.405
12.405
Retention Levels
0.405
4.405
8.405
12.405
Retention Levels
Figure 2. Comparison of rankings under the standard deviation premium principle when u = 5 and
t = 1.
These results are consistent with the expected value premium principle case, which was shown in
Section 4.1.2. TOPSIS-E and M-TOPSIS give the same optimal retention level for each pair of initial
surplus and time horizon. Since the variance of the insurer’s risk is involved in the standard deviation
premium principle, the necessity of reinsurance is higher so that the retention level is smaller than the
expected value premium principle case. In TOPSIS-Mahalanobis, the effects of the existence of the
relationship between the criteria are clearly seen in the standard deviation premium principle. Higher
optimal levels are obtained in the standard deviation premium principle due to the higher covariances.
The revised VIKOR method produces the same optimal retention levels for both premium principles.
The covariance matrices of normalized attributes for t = 1, 5, 10 are given in Table 7.
J. Risk Financial Manag. 2017, 10, 4
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Table 6. Optimal retention level under the standard deviation premium principle.
Optimal Retention Levels
u=1
u=5
u = 10
TOPSIS-E
M-TOPSIS
1.0055
0.8055
0.8055
0.7055
0.8055
0.8055
0.7055
1.2055
1.0055
1.0055
0.9055
0.8055
0.8055
0.6055
1.2055
1.0055
1.0055
0.9055
0.8055
0.9055
0.7055
1.0055
0.8055
0.8055
0.7055
0.8055
0.8055
0.7055
1.2055
1.0055
1.0055
0.9055
0.8055
0.8055
0.6055
1.2055
1.0055
1.0055
0.9055
0.8055
0.9055
0.7055
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
t=1
t=5
t = 10
t = 15
t = 20
t = 25
t = 50
TOPSIS-Mahalanobis VIKOR
2.9055
2.9055
2.9055
2.7055
2.7055
2.7055
2.3055
1.5055
1.8055
1.7055
1.6055
1.3055
1.6055
1.2055
1.5055
1.1055
1.3055
1.5055
1.5055
2.2055
1.1055
R-VIKOR
1.2055
1.1055
1.1055
1.1055
1.1055
1.1055
1.1055
1.4055
1.3055
1.4055
1.3055
1.3055
1.3055
1.2055
1.4055
1.3055
1.4055
1.4055
1.4055
1.4055
1.3055
2.8055
2.9055
2.9055
2.9055
2.9055
2.9055
2.9055
14.9055
14.9055
14.9055
14.9055
14.9055
14.9055
14.9055
29.9055
29.9055
29.9055
29.9055
29.9055
29.9055
29.9055
Table 7. Covariance matrices of normalized attributes for t = 1, 5, 10 when u = 5.
Covariance Matrices
t=1
Var ( X I )
ES p
EP
ψ(u, t)
Var ( X I )
5.69×10−4
8.12×10−5
−3.64×10−6
3.88×10−4
ES0.95
8.12×10−5
1.88×10−5
−6.47×10−7
5.75×10−5
EP
−3.64×10−6
−6.47×10−7
2.61×10−8
−2.52×10−6
ψ(u, t)
3.88×10−4
5.75×10−5
−2.52×10−6
2.67×10−4
t=5
Var ( X I )
ES0.95
EP
ψ(u, t)
Var ( X I )
5.69×10−4
1.27×10−4
−2.85×10−5
3.88×10−4
ES0.95
1.27×10−4
3.62×10−5
−7.71×10−6
8.84×10−5
EP
−2.85×10−5
−7.71×10−6
1.67×10−6
−1.98×10−5
ψ(u, t)
3.88×10−4
8.84×10−5
−1.98×10−5
2.67×10−4
t = 10
Var ( X I )
ES0.95
EP
ψ(u, t)
Var ( X I )
5.69×10−4
1.53×10−4
−5.15×10−5
3.88×10−4
ES0.95
1.53×10−4
4.92×10−5
−1.67×10−5
1.06×10−4
EP
−5.15×10−5
−1.67×10−5
5.67×10−6
−3.59×10−5
ψ(u, t)
3.88×10−4
1.06×10−4
−3.59×10−5
2.67×10−4
5. Conclusions
In this study, we have calculated the optimal retention levels by using four competitive criteria
under the assumption that the aggregate claims have a compound Poisson process. We have obtained
sets of alternatives that comprise the pair of initial surplus and retention level. Then, we have calculated
survival probability, expected profit, variance and expected shortfall with regard to these pairs.
We have determined the optimal retention level that makes the survival probability and expected
profit maximum, whereas variance and the expected shortfall minimum. In the decision making
process, we have used two major MADM methods: TOPSIS and VIKOR, as well as their extended
versions. We have compared the findings for different initial surpluses, time horizons for both expected
value and standard deviation premium principles.
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Based on the results, it can be concluded that the TOPSIS method with Euclidean distance and
modified TOPSIS give the same results, and the VIKOR method produces very close optimal levels to
TOPSIS-Mahalanobis. However, from the research that has been carried out, it might be concluded
that since the relationship between the criteria is too small, the distance measure does not cause
any significant effect on determining the optimal retention level. Hence, the TOPSIS method with
Euclidean or Mahalanobis distance produces very close optimal retention levels. On the other hand,
the premium principle technique has a vital role in optimal reinsurance. Since the variance is used
in the standard deviation premium principle, lower optimal values are observed in TOPSIS-E and
M-TOPSIS. On the other hand, the higher optimal levels are obtained when the relationship between
the criteria is included.
The proposed method is applicable to different claim amounts and claim number distributions, as
well as to other MADM methods.
Acknowledgments: The authors thank two anonymous referees for their helpful comments and suggestions.
Author Contributions: The authors contributed equally to this work.
Conflicts of Interest: The authors declare no conflict of interest.
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